Local Interpretable Model-Agnostic Explanations (LIME) are a well-known approach to provide local interpretability to Machine Learning models. LIME uses an exponential smoothing kernel based on the kernel width value, which defines the width of the local neighbourhood. In this paper, we study the influence of the distances for these local explanations, and we explore the choice of kernel width to guarantee a fair performance comparison between the distances.
The gap between the recent development of AI models and their social use has led the scientific
community to develop a new research area called Explainable Artificial Intelligence (XAI)
(see, e.g., [
One of the key points in these local explanations is the meaning of local. In this way, the
sense of proximity among points in the training dataset of the model plays a central role in
local explanations, and hence the choice of the distance considered is crucial in order to find
plausible explanations. In this paper, we consider one of the most widely used XAI methods,
the well-known method LIME [
LIME is a XAI model to provide interpretability for individual instances1. Each local explanation
is a set of feature-value pairs that determine which features provide a greater contribution to the
prediction, together with a numeric value that quantifies this contribution. In the literature, we
can find several studies of LIME focussing on diferent improvements on the original algorithm as
[
The reliability of the Euclidean distance to capture the intuition of proximity in high-dimensional metrics spaces has been widely studied. For example, in [18] the authors explain how using traditional metrics (, ) = ∑︀=1 | − |1/ in high-dimensionality problems leads to a loss of the notion of proximity. This is a major concern in problems where proximity plays an important role. LIME uses a metric to measure the distance between binary vectors whose components are 0’s and 1’s. In this context, the Hamming distance can be expressed as Hamming(, ) = ∑︀
=1 | − | which is exactly 1. In the case of binary vectors, the only diference between applying the Hamming or Euclidean distance is the square root, given that 12 = 1 and 02 = 0. Furthermore, the maximum in the Hamming distance between two binary sequences in R is , but the maximum distance between two binary sequences in R in the Euclidean distance is √, as one is the square root of the other. In the LIME implementation, the default value given to the kernel width is 0.75 · √, which can be seen as 75% of the maximum 1Along this paper, we refer exclusively to explanations of tabular data. value given by Euclidean distances over an n-dimensional space. Using the same idea, we define a custom kernel width for the Manhattan distance as 0.75 · . Analogously, a kernel width comparable to the Euclidean metric performance (using the custom definition) for each is given by = 0.75 · 1/.
In this experiment, we study the appropriate kernel width to be used depending on the metric.
Firstly, diferent synthetic data sets composed of 500 samples of diferent dimensions (from
10 to 40 with a step of 10) were generated. Secondly, for each dimension, a Random Forest
classifier was trained using 90% of the dataset as a training set. Thirdly, LIME was used to
explain the remaining 10% of the data set (used as a test set) using all the attributes available.
The metric is computed for the coeficient stability [
In this second experiment, we compare the explanations obtained using LIME using both distances. The explanations provided by LIME are an ordered list of the attributes based on its contribution. The dataset used is the UCI ML Breast Cancer Wisconsing (Diagnostic) dataset composed of 569 samples of dimension 30 for binary classification. The dataset was split into training and test set in a proportion 90 − 10 and a Random Forest classifier was trained. Then, explanations were computed for the remaining 10%, obtaining a vector of the 30 attributes ordered by importance. This was done using the predefined kernel width for the Euclidean distance, and both the predefined and the proposed kernel width for the Hamming distance. The vectors were compared as follows: Given two vectors , with length the number of
CSI
Variance 0.1 8.53 0.12 features in the dataset, for each feature we compute the diference between the coordinates of the feature in and . Finally, the mean of these diferences is computed. This provides a similarity measure in the order of importance of the attributes. In Table 1 (left), the coeficients CSI are shown, depicting the stability reached by LIME using the diferent kernel widths. Let us remark that the conclusions of Experiment 1 are also achieved. We can see that the high values are reached for the Euclidean distance using the predefined kernel width and the Manhattan distance using the proposed one. In Table 1 (right), the similarity measure between the order of the features computed using LIME is provided. Therefore, it is shown that both distances have similar performance, providing similar relevance of the same features.
In the foundations of Machine Learning, it is well-known that the Euclidean distance loses the proximity notion for high dimensions, while the Hamming distance performs better in that context. However, this fact is not considered in the standard use of LIME and hence, undesirable explanations can be obtained. In this paper, we have studied the relationship between the Euclidean and the Hamming distances in this context, concluding its similarity. Experimentally, we have shown that the stability of the LIME algorithm converges and that the resultant order of importance of the features is similar when using the Euclidean and the Hamming distance if the kernel width is adapted to the chosen distance.
Partially supported by REXASI-PRO H-EU project, call HORIZON-CL4-2021-HUMAN-01-01, Grant agreement no. 101070028, and national projects PID2019-107339GB-I00 and TED2021129438B-I00 funded by MCIN/AEI/ 10.13039/501100011033 and NextGenerationEU/PRTR. The content reflects the views of the authors only.
2008.11092. [17] G. Visani, E. Bagli, F. Chesani, A. Poluzzi, D. Capuzzo, Statistical stability indices for LIME: Obtaining reliable explanations for machine learning models, Journal of the Operational Research Society 73 (2021) 91–101. URL: https://doi.org/10.1080%2F01605682.2020.1865846. doi:10.1080/01605682.2020.1865846. [18] C. C. Aggarwal, A. Hinneburg, D. A. Keim, On the surprising behavior of distance metrics in high dimensional space, in: J. Van den Bussche, V. Vianu (Eds.), Database Theory — ICDT 2001, Springer Berlin Heidelberg, Berlin, Heidelberg, 2001, pp. 420–434.